Areas of Figures Broken into Rectangles

Date: 10/17/2001 at 20:16:19
From: Rachel
Subject: Area of figures
On my sheet it tell me to "Calculate the area of each figure. First
divide the figure into rectangles and squares."
I have divided them into the rectangles and squares, but I don't know
how to find the area. Please help!
Thanks,
Rachel

Date: 10/18/2001 at 11:03:57
From: Doctor Ian
Subject: Re: Area of figures
Hi Rachel,
Once you've divided something into rectangles,
+------+-------------+
| A | |
| | |
+------+ B |
| |
| +-------------------+
| | C |
+-------------+-------------------+
you can find the total area by adding up the areas of the individual
rectangles.
If you don't see why this is true, you might be a little confused
about the meaning of 'area'. If that's the case, you might want to
look at this answer from the Dr. Math archives:
Area and Perimeter
http://mathforum.org/dr.math/problems/jessica.5.1.01.html
Think of it this way. Suppose I need to paint the shape I've drawn
above. To find out the total amount of paint I need, I would add up
the amounts needed to paint the individual rectangles. Does that make
sense?
So, how do you find the area of a single rectangle? You find two
adjacent sides (that is, two sides that touch at a corner) and
multiply their lengths:
3
+--------+
| | 2 area = 3 * 2 = 6
| |
+--------+
Note that a square is just a special kind of rectangle, in which each
pair of adjacent sides has the same length. So the formula still
works.
You may be able to divide the same shape into rectangles in more than
one way, for example:
+--------------------+
| |
| |
+------+ |
| |
| +-------------------+
| |
+---------------------------------+
+------+-------------+
| A | |
| | |
+------+ B |
| |
| +-------------------+
| | C |
+-------------+-------------------+
+------+-------------+
| A |
| |
+------+-------------+
| |
| +-------------------+
| B | C |
+-------------+-------------------+
+--------------------+
| A |
| |
+------+-------------+
| B |
+-------------+-------------------+
| C |
+-------------+-------------------+
+------------+-------+
| A | B |
| | |
+------+-------------+
| C |
+---------+---+---------+---------+
| D | E | F | G |
+---------+---+---------+---------+
In each case, so long as the individual rectangles cover the whole
shape, you'll get the same total area, no matter which rectangles you
choose. (Again, if you think of area in terms of paint, it should be
clear why this has to be the case.)
Generally, you want to choose rectangles where you'll be told (or be
able to figure out) the lengths. Often the difference between an easy
solution and a hard solution is the way you decide to break a large
shape into smaller shapes.
It's really not that different from breaking up any large quantity
into smaller quantities that are easy to find. For example, suppose
you need to know the distance from Long Beach, CA to Chesterton, IN.
You might not be able to find a table of distances that will give you
that distance directly; but perhaps you can find three other tables
that give you the distance from Long Beach to Los Angeles, the
distance from Los Angeles to Chicago, and the distance from Chicago to
Chesterton. Then you can add those three distances to get the total
distance (assuming that you plan to drive through Los Angeles and
Chicago on the way).
Does this help? Write back if you'd like to talk about this some
more, or if you have any other questions.
- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/